On Quantum Wave Groups
In this article the "spinning spiral" model for the complex quantum wave function Y is investigated. The associated particle is a free electron moving in the positive x direction at a most probable speed of vo. At any given x a complex plane is assumed to exist perpendicular to the x-axis. The Imaginary axis of this plane is parallel to the y-axis, and the Real axis is parallel to the z-axis. Representations of Y in any of these complex planes will be looking down the x-axis in the positive x direction (See Fig. 1_1).
Complex Plane Parallel to the yz Plane
The electron is considered to be localized which, according to the Heisenberg Uncertainty Principle, implies that there is a finite probability it could be found to have any of a range of momenta p. The localization is defined by
For purposes of computation, the momenta are assumed to be discrete, say pi where i ranges from 1 to N. Each momentum in the set has an associated Yi with its own wave number (ki) and angular frequency (wi).
As suggested in a previous article, each Yi spins CW with increasing time and spirals CCW with increasing x. The angular rate at which Yi spins at any given x is defined by the deBroglie relation
And the rate at which Yi spirals at any given time is defined by
For the localized particle, Y is the sum of the set of Yi. The magnitude of each Yi is 1/N, thus assuring that the magnitude of the resultant Y is never greater than unity. It is noteworthy that the individual Yi need not be weighted in order to obtain a wave packet (although this is often done for purposes of fourier analysis, etc.). For graphic purposes, YY* (which is always real) is plotted vs. x. The general goal is to determine (a) if the Yi sum to a quasi-gaussian group, and (b) if the group velocity equals vo (the most probable velocity).
2. The Ranges of p and x.
We assume that, at t=0, all the constituent Yi are Real and are in phase at x=0. Since the wave numbers vary, the constituent Yi will fall out of phase as we move away from the origin. The uncertainty in x is assumed to be Dx=1E-6 meters (wavelength of visible light). The most probable momentum is specified to be po=.01mc, where m is the electronís mass and c is the speed of light. The uncertainty in p is Dp=h/Dx, and p ranges from po-Dp/2 to po+Dp/2.
3. Plots of YY*.
We begin by computing the resultant Y at time t=0 and over the range -2Dx<x<4Dx. Fig. 3_1 plots YY*(x) at t=0. Note the well-defined group and (since N is finite) the side lobes.
Fig. 3_2 plots YY*(x) for t=2Dx/vo. Note how the group has propagated to the right a distance of vot. That is, the constituent Yi are now all in phase at x= vot.